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Analysis of fatigue crack growth under mixed mode (I + II) loading conditions in rail steel using CTS specimen
Journal Pre-proofs Analysis of fatigue crack growth under mixed mode (I+II) loading conditions in rail steel using CTS specimen Grzegorz Lesiuk, Michał Smolnicki, Rafał Mech, Anna Zięty, Cristiano Fragassa PII: DOI: Reference:
S1350-6307(19)31640-1 https://doi.org/10.1016/j.engfailanal.2019.104354 EFA 104354
To appear in:
Engineering Failure Analysis
Received Date: Revised Date: Accepted Date:
4 November 2019 8 December 2019 16 December 2019
Please cite this article as: Lesiuk, G., Smolnicki, M., Mech, R., Zięty, A., Fragassa, C., Analysis of fatigue crack growth under mixed mode (I+II) loading conditions in rail steel using CTS specimen, Engineering Failure Analysis (2019), doi: https://doi.org/10.1016/j.engfailanal.2019.104354
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© 2019 Published by Elsevier Ltd.
Analysis of fatigue crack growth under mixed mode (I+II) loading conditions in rail steel using CTS specimen
Grzegorz Lesiuk1*, Michał Smolnicki 1, Rafał Mech1, Anna Zięty1, Cristiano Fragassa2 1
Wroclaw University of Science and Technology, Faculty of Mechanical Engineering, Department of Mechanics, Materials Science and Engineering, Smoluchowskiego 25, PL50-370 Wrocław; Poland
2
University of Bologna, Department of Industrial Engineering, Viale Risorgimento 6, 40133 Bologna, Italy
* Corresponding Author: [email protected] [G.L.] Abstract: The paper is focused on the investigation of fatigue crack growth in rail steel. Mode I and its combination with mode II are one of the frequent reason for rail failures. Therefore, mode I and mixed-mode (I+II) experiments were carried out in order to obtain the fatigue crack growth curves for typical rail steel. In experimental campaign were used CTS (Compact Tension Shear) specimens. The Stress Intensity Factors (SIF) were determined for all load cases using Finite Element Method (FEM) with developed and automated Authors’ tool ABAQUS environment. During the experimental and numerical study, the impact of different elastic mixity level on fatigue crack growth rates and fatigue crack growth directions has been discussed. As it is noticeable the decrease of the mode I domination (under mixed-mode loading condition) caused an increase of the specimen fatigue lifetime. This observation was also discussed in the light of the detailed fatigue crack paths study using Scanning Electron Microscope (SEM). Keywords: fatigue fracture mechanics, fatigue crack path study, mixed-mode I+II fatigue crack growth, CTS, SIF, FEM, rail steel
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1. Introduction The safety of a steel rail structure is often determined by the maintenance of the rail system and the prediction of its failure based on the fracture process. The mechanics of train wheel-rail movement causes a specific configuration of stress state in rail structure so-called Rolling Contact Fatigue (RCF). The cyclic nature of loading combined with multiaxial stress state in rail determines the specific crack initiation point and fatigue crack paths during cyclic loading. The nature of the type of crack morphology is well reported in the works [1-2]. On the other hand, the fatigue crack growth properties are still required in several numerical approaches in fatigue lifetime prediction of structural elements and components like rails. Also, fatigue crack growth rate (FCGR) parameters are often one of the major criteria for evaluation of the rail steel quality (i.e. pearlitic, bainitic etc.). Recently, Nejad et al. [3] and Christodoulou [4] studied the fatigue crack growth behaviour under mode I of rail steel 900A regarding the influence of stress ratios R=0.1, 0.2, 0.3. However, in some cases, the mixedmode (I+II) fatigue crack behaviour is required in fatigue analysis of the structural integrity. Therefore the Authors decided to investigate the same class 900A rail steel behaviour under mixed-mode (I+II) in order to investigate real behaviour of mixed-mode fatigue crack growth in rail steel class 900A. As it is well known, in some cases, due to material flaws and heterogeneity with a combination of the complex stress state, cracks propagate under mixed-mode loading conditions. The problem needs an individual approach to research work in comparison to the standard problem of one mode fracture, which is already well elaborated. Most of the theoretical works are focused on crack initiation and propagation criteria [5-8]. Between many others, are noteworthy: Maximum Tangential Stress (MTS) [6], Maximum Energy Release Rate (MERR)[7] and Minimum Strain Energy Density (SED)[8]. For the validation of such criteria, multiaxial loading tests are required, as widely described in [9-11] with mode I+II using cruciform specimens [9, 12, 13]. From the practical point of view, the mixed-mode fracture mechanics is mostly studied using uniaxial tensile machines. At present, there are different types of specimens, which can be used to test fracture under mixed-mode (I+II) loading using the uniaxial tensile machine. For brittle materials are often used disc-type specimens such as Centre Cracked Circular Disc (CCCD) [11] and Semi-Circular Bend (SCB) [14]. Other popular types of specimens are Single Edge Crack (SEC)[15], Single Edge Notched Beam[15]. Another group of specimens is based on the extended concept of CT (Compact Tension) specimen; Arcan (“butterfly-shaped”) [16, 17] and CTS (CTS) [18, 19] specimens. As it is observed, this type of specimen is most frequently used in material
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characterisation under mixed-mode (I+II) condition. For static load conditions, SIF (SIF) solutions are available in the literature [20]. However, some of the significant problems in engineering practice are related to fatigue and fatigue fracture. In those cases, the closed-form SIF solutions are invalid [21-22]. As an alternative, a FEM (FEM) methodology can be used for the evaluation of SIFs [23-26]. Based on this, the numerical support of the experimental part is an essential contribution for fatigue crack growth analysis under mixed-mode loading condition of rail steel. According to the proposed methodology and obtained experimental results, it can help avoid critical situations (using a damage tolerance approach) of railway lines. 2. Material and Methods 2.1. Material investigations As an object of Authors’ interest, steel coming from the rail line (profile UIC60) was selected. For the tests, the unserviceable rail was taken directly from the manufacturer. Samples for examination were cut out under cooling with liquid in order to avoid microstructural changes. The microscopic observations were performed using light microscopy (LM) in the non-etched and etched state (5%HNO3). The microstructure of the tested steel is shown in Figure 1. The chemical composition of the material was spectrally analysed. The results are summarized in Table 1. Table 1. Chemical composition (in % by weight) of delivered rail Material
C
Mn
Si
P
S
Cr
Ni
Mo
Fe
tested rail steel
0.721
0.873
0.256
0.012
0.005
0.053
0.032
0.011
bal.
Static properties (UTS – ultimate tensile strength, spl – yield stress, A5 – elongation at break, HV – Vickers Hardness) for selected material are collected in Table 2 (mean value of five results). The static tensile test was performed following the PN-EN ISO 6892-1:201609 standard using round specimen with diameter Ød=6mm extracted from the rail. Stressstrain curves collected from the static tensile test are presented in Figure 2. According to the chemical composition, the microstructure of the tested material corresponds to the perlite structure of steel, typical for railway rails.
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Table 2. Static mechanical properties of analysed metallic materials Material Extracted rail steel
UTS
spl/s0.2
A5
HV
(MPa)
(MPa)
(%)
(-)
998
481
14.5
258
Figure 1. The pearlitic microstructure of the extracted material from UIC60 rail; a) representative image with non-metallic inclusions chains in the non-etched state (200x) b) typical microstructure of tested steel (200x), c) enlarged pearlite structure with ferrite grains (400x), d) typical fatigue crack path in rail steel (100x)
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1000
(MPa)
800 600 400 200 0 0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
Figure 2. Monotonic stress-strain curves for tested rail steel 2.2. Mode I and mixed-mode (I+II) fatigue crack growth – theoretical and experimental aspects In 1957 Williams [27], proposed an asymptotic expansion of the stress field in the area surrounding the tip of the sharp crack. Original formula (Equation 1), reporting the stresses ߪ in the function of distance from the crack tip r, Young Modulus E, Poisson’s coefficient ߥǡstrains ߜ ǡand angle η, is presented below: ߪ ൌ
ܭூ
ξʹߨݎ
݂ூ ሺߟሻ
ܭூூ
ξʹߨݎ
݂ூூ ሺߟሻ
ܭூூூ
ξʹߨݎ
݂ூூூ ሺߟሻ ܶߜଵ ߜଵ
(1)
ሺ ܶݒ ߝܧଷଷ ሻߜଷ ߜଷ ܱ൫ξݎ൯
As is it well known, that here exist SIFs for three modes: KI, KII, KIII as well as on quantity T – called T – stresses. The latter one is the transverse component of stress and was first proposed by Irwin [28] to deal with problems using Westergaard solution [29] to a uniaxial type of loading. Gupta et al. [30] gathered information about the influence of the T-stress parameter. This approach is also reflected in various approaches in mixed mode fatigue fracture description as in [31-33]. However, T-stress influence is still investigated and needs intensively study on in several types of specimens in order to standardise testing procedures. On the other hand, during mode I and mixed mode-fatigue crack growth, the most accepted approach is based on the DK approach. It is also preferred in most widely used American Standard - ASTM E647 devoted to the description of testing method, but only for mode I. In case of the mixed-mode loading type (I+II) the CTS Specimen is most widely used.
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This specimen was designed and investigated by Richard [18, 19, 33, 34]. It is worth to note that this specimen needs an additional fixture for clamping on the uniaxial tensile machine. The general concept and an exemplary measurement stand is shown in Figure 3.
Figure 3. CTS specimen mixed-mode loading configuration; (a) general concept; (b) definition of θ- angle; (c) experimental measurement stand; 1 – load cell, 2 – grips, 3 – specimen holder, 4 – CTS specimen, 5, 6 - lightening system, 7 stereoscopic microscope, 8 – digital camera By changing the load direction θ it is possible to create different mixed-mode loading conditions; for ߠ ൌ Ͳιis pure mode I (so ܭூூ ൌ Ͳ), for ߠ ൌ ͻͲι - pure mode II (so ܭூ ൌ Ͳ) and between these values of θ, there are mixed-mode loading conditions. Figure 4 presents
the main dimensions of CTS specimen [15]. Richard proposed closed-form solutions for SIFs using fundamental geometrical aspects of CTS specimen like a – crack length, W – specimen width, t – specimen thickness (see Equation 2 and Equation 3) [34]. ܽ ͲǤʹ ʹǤͷ ቀܹ െ ܽቁ ܨή ξߨܽ ή ߠ ܭூ ൌ ଶ ܽ ඩ ܹ ݐቀͳ െ ܹ ቁ ͳ ͲǤͷͷ ቀ ܽ ቁ െ ͲǤͺͺ ቀ ܽ ቁ ܹെܽ ܹെܽ
ܽ െͲǤʹ͵ ͳǤͶ ቀܹ െ ܽቁ ܨή ξߨܽ ή ߠ ܭூூ ൌ ଶ ܽ ඩ ܹ ݐቀͳ െ ቁ ͳ െ ͲǤ ቀ ܽ ቁ െ ʹǤͲͺ ቀ ܽ ቁ ܹ ܹെܽ ܹെܽ
(2)
(3)
However, the cited equations are only valid for stationary cracks.
This type of CTS-specimen is also suitable for cyclic loading. In the experimental campaign was used MTS 809 servo-hydraulic machine (Figure 3) equipped with a digital camera for capturing the crack tip on the CTS specimen (schematically drawn in Figure 4). Three different load angles were considered θ = 0°, 30°,45°. Before fatigue crack growth test,
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the precracking procedure was involved by cyclic loading with ܨ௫ ൌ ͷǤͷ݇ܰ and ܴ ൌ
ͲǤͳduring 150,000 cycles and with ୫ୟ୶ ൌ ͶǤͷ and ൌ ͲǤͳ during remaining 50,000
cycles. After precracking procedure, the test was performed using a constant load amplitude method. During experiment Fmax=8 kN, f=10 Hz and R=0.1 were kept.
Figure 4. Geometric scheme of the exemplary CTS specimen used in the experimental and numerical campaign (all dimensions in mm, thickness t=8mm) In order to estimate fatigue crack growth behaviour under the mode I and mixed-mode (I+II), the kinetic fatigue fracture diagrams (KFFD) were constructed. Directly from the experiment, the fatigue crack lengths and number of cycles to failure were registered. Using an automated digital imaging system (Figure 3), the crack tip was detected and recorded in the local coordinate system. It is located at the end of precrack (mode I) length – Figure 4. Fatigue crack growth curves are shown in Figure 5.
Figure 4. Exemplary CTS specimen after the test (load angle 45 degrees) with real crack path and the definition of the coordinate system for numerical analysis
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Figure 5. Fatigue crack growth curves for tested rail steel After fatigue experiments, the mixed-mode (I+II) crack initiation angles ψ0 were measured using the microscope. All results are presented in Figure 6.
Figure 6. Crack initiation angles in CTS specimens after fatigue crack growth test The detailed images of fracture surface for mode I and mixed-mode (I+II, load angle 45 degrees) are presented in Figs 7-10. For mode I, the fracture surface is mostly shaped by the typical mode I, the dominant tensile mechanism of fracture. It is noticeable in enlarged microscopic view (Figure 7) the numerous secondary cracks characteristic for higher mode I, tensile stress level. This transgranular type of fracture also shows pearlite colonies and lamellar spacing in steel grains. This type of fracture is dominant along the fatigue crack path. Only last, overloaded part (Figure 8) of the specimen is shaped by brittle, transgranular facets with secondary cracks.
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Figure 7. Initial crack path and fracture surface (mode I) of tested material – macro view and marked by frame view on the fracture surface
Figure 8. Final crack path (3mm) and overloaded fracture surface (mode I) of tested material– macro view and marked by frame view on the fracture surface The fracture surface of the specimen subjected to mixed-mode (I+II) loading type is different – Figure 8. At the initial region, here is a noticeable sliding type of fatigue crack growth (magnified view) – it might be caused by relatively high KII/KI ratio on the initial crack length. However, in the last stage of fatigue crack growth (Figure 10), it is a noticeable dominant mode I, tensile type of fatigue crack growth mechanism, similar to pure mode I.
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Figure 9. Initial crack path and fracture surface (mode I+II, loading angle 45 deg.) of tested material – macro view and marked by frame view on the fracture surface
Figure 10. Final crack path (3mm) and overloaded fracture surface (mode I+II, loading angle 45 deg.) of tested material– macro view and marked by frame view on the fracture surface 3. Numerical study of the CTS specimen – SIF estimation In order to construct fatigue crack growth rate diagrams, it is necessary to calculate SIFs. The SIFs solutions provided by Richard are valid only for stationary cracks, which is reported in papers of Richard [33,34] and proven in many applications in [20,21]. On the other hand, there is a strong need to elaborate generalised model in a closed-form solution for CTS specimen most widely used in fatigue fracture mechanics. According to the generalised form of SIF solution, dimensionless quantities YI, YII are used below as independent from of force value and stress level: ܻூ ൌ
ܭூ
ɐඥߨܽ
ൌ
ܭூ ή ܹݐ
ή ɐඥߨܽ
(4)
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ܻூூ ൌ
ܭூூ
ɐඥߨܽ
ൌ
ܭூூ ή ܹݐ
ή ɐඥߨܽ
(5)
In the classic approach where the stationary crack is investigated aeff means crack length which is equal to the projection of this crack on the x-axis. However, during fracture under mixed-mode loading, crack tends to deviate from a straight line. Thus, two possible values of aeff are considered: total length of notch depth, precrack and crack as well as their projection on the x-axis. 3.1. Automatic generation of the numerical model Since the multiplicity and the relevance of parameters affecting the tests in the case of mixed-mode loading conditions, automation in the simulation process is strictly necessary for efficient use of the CTS specimen method, For instance, even dimensionless ܻூ ǡ ܻூூ still
depends on a few different quantities.
In work [35] authors using Buckingham π theorem show that this is possible to reduce ௫
௬
the set of quantities defining dimensionless SIFs to ቄௐ ǡ ௐ ǡ ߚǡ ߙ െ ߚቅwhere ߙ – crack angle and ߚ – force direction angle. Of course, there are other sets that can fulfil the same function. ௫
Due to the character of the prepared script in this work set ሼௐ ǡ ௐ ǡ ߙǡ ߚ } is used. This
approach is more complicated for finding dependencies for a specific location of the crack tip, but is more natural, when using with experimental research.
Simulia Abaqus environment allows taking advantage of scripting with the reduction of the analysis complexity. Based on Abaqus Scripting User Guide [36] the Abaqus Scripting Interface is an application programming interface (API), which gives the possibility to access to the models and data operating by Abaqus. The Abaqus Scripting Interface is based on Python language, thus it is possible to interact with Abaqus objects by using Python scripts. The User has different possibilities, inter alia, ·
creation and changing Abaqus object i.e. parts, sections, materials, loads, boundary conditions and steps,
·
management of Abaqus jobs, including submit,
·
management of Abaqus output database. Using Abaqus Scripting Interface a script to simulate fracture in CTS specimen was
prepared. In Figure 11 flowchart of this script is presented.
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Figure 11. Flowchart of the script for fatigue fracture analysis of CTS specimen First, input from the user about specimen geometry is required, crack parameters and initial notch/precrack. Based on this information, the geometry model is created (eventually included data from the previous iteration, if present), then crack is included in the model via partitioning. In the next few steps, the numerical model is created i.e. material, assembly, boundary condition, finite element mesh and history output are defined. In the purpose of obtaining fracture data crack region is defined by using the Contour Integral Method. Finally, the work submission is automatically realized and, after results are provided, data necessary to calculate the new position of crack and fracture parameters are obtained. Part of the data also serves as input for the next iteration. The whole process is continued until the specimen is not fully broken into two parts. Result of one run of presented script is a set of models and result files for the subsequent increments of the crack for the user-defined geometry and other initial values like force action direction. In addition to the above text file with aggregated data is generated. Using loops, it
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is possible to automatically generate a model for a user-defined list of parameters i.e. for 12 different initial precrack lengths. Consequently, the whole procedure of obtaining requested data is automatized. According to the main goal of this paper, the CTS specimen was numerically investigated in order to extract essential fracture mechanics parameters for FCGR description in rail steel. In Figure 12, a scheme of the specimen used in the experimental and numerical campaign is presented. The main dimensions of this type of specimen are expressed by: the length of precrack – ܽǡ the spacing of pins –ܾ, thickness ݐǡ and width ݓare previously
used in some formulas for calculating SIFs and will be discussed in the latter part of this work.
Figure 12. Scheme of CTS specimen: length of precrack – a , pins spacing – b, thickness t, and width W For calculating SIFs the Contour Integral method (as detailed later in the article) was involved. Material data specified in Abaqus simulations was following; Young Modulus ܧൌ ʹͳͳͲͲͲܽܲܯand Poisson’s ratio ߥ ൌ ͲǤ͵Ǥ
Several different ways to apply boundary conditions are proposed in the literature in the
case of CTS specimens. In Figure 13a (later referenced as Boundary Conditions type 1 – BC1) and Figure 13b (later referenced as Boundary Conditions type 2 – BC2), two popular methods are presented [37-39]. In the first case, the supports on the bottom part of the model generate reaction forces which are equal to forces applied in these points in the second case. These two propositions were compared and checked via preliminary FEM simulations. In the presented study, the Richard [34] formula (cf. Equation 2 and Equation 3) was used to evaluate these results in the basic case when there is no crack propagation.
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Comparison of SIFs for the standard dimension (as described in subchapter geometric model) and ߠ ൌ ͵Ͳιis shown below in Table 3.
(a)
(b)
Figure 13. (a) Boundary conditions type 1. Forces on one side of the specimen and fixing on the other side; (b) Boundary conditions type 2. Forces on both sides of the specimen. Table 3. Comparison between analytical solutions and results from different types of BC BC – type I ܭூ ሾ ܽܲܯή ݉݉Ǥହ ]
ܭூூ ሾ ܽܲܯή ݉݉Ǥହ ]
400.53 -116.64
Richard
BC – type II ȟ ൌ ͲǤ͵Ψ ȟ ൌ ͳǤͷΨ
405.06 -72.79
Solution ȟ ൌ ͳǤͶͻΨ
ȟ ൌ ͵ͺǤΨ
399.11 -118.73
According to data presented in Table 3. first type of boundary conditions was selected. Forces ܨଵ ǡ ܨଶ ǡ ܨଷ were calculated based on two conditions: first - the resultant force has the
desirable direction of action ߠ and second -resultant momentum of forces is equal 0, so there
is no bending. Forces ܨଵ ǡ ܨଶ ܽ݊݀ܨଷ which meet these requirements were described in equations 6, 7 and 8. On the other side of specimen degree of freedom was taken as it is shown in Figure 13a. ܿ ܨଵ ൌ ܨή ቀͲǤͷ ߙ ߙቁ ܾ ܨଶ ൌ ܨή ߙ
(6) (7)
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ܿ ܨଷ ൌ ܨή ቀͲǤͷ ߙ െ ߙቁ ܾ
(8)
Due to two-dimensional modelling space, the force applied to the specimen ܨmust be
normalised via division by the thickness of the specimen ܨௗ ൌ ܨȀݐ. Both forces and boundary conditions are applied to the pins represented by reference points in Abaqus. Kinematic coupling (all degree of freedom) is used to model these conditions. Discrete model, as well as previous elements of the numerical model, is generated in the automatized process. The size of the elements is one of the parameters which can be regulated. The influence of the element size on the obtained exemplary SIF values for CTS specimen is shown in Fig. 14.
Figure 14. KI and KII values vs. finite element size – load angle 45°, mixed-mode fatigue crack length 5 mm According to the exemplary data in Fig. 14, it is noticeable, that for the smaller than 0.5 mm finite element sizes, there is a significant change in the obtained SIF values. In the presented case, the mesh size estimated on the level about 0.5 mm corresponds approximately to a 1/80 specimen width. During simulations, mostly linear quadrilateral type CPS4R (4node bilinear plane stress quadrilateral with reduced integration as well as hourglass control) were used. The general view of the discrete model is presented in Figure 15a. Regarding the fact that the J-integral and T-stresses are calculated, regular circular layers of finite elements around the crack tip were applied - Figure 15b. In another region, the more coarse mesh has been applied.
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(b) (a) Figure 15. A discrete model of the CTS specimen (a) Mesh on the whole specimen, size of element circa equal width of a specimen divided by 80; (b) Mesh in the crack region - 10 contours for calculating SIFs, J-Integral and T-stress There are few different ways to obtain SIFs, J-Integral and T-stresses in Abaqus environments such as eXtended FEM and Contour Integral. First of them can be easily applied in more complicated cases. In the case of the two-dimensional model, the second of them is preferred. This procedure may cause problems with the presence of a singularity near the crack tip. In order to overcome this, a standard procedure was applied. Nodes at the crack tip were translated by 0.25 of element length. Besides, the collapsed element side (single node) method was selected for dealing with the degenerate element. Here, it is required to deal with the problem of singularity in the tip of the sharp crack [40]. The direction of the crack extension was defined based on the crack increase from the previous step (or based on the direction of precrack for the first simulation). The exemplary definition is shown in Figure 16.
Figure 16. Definition of the crack extension direction
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3.2. Results of numerical simulations To determine the influence of carried
out
for
the
ௐ
following
ratio as well as force direction ߠ simulations were distance
of
notch
and
precrack:
ܽ א
ሼͳʹǡ ͳǡ ʹͲǡ ʹͶǡ ʹͺǡ ͵ʹሽand following force direction: ߠ אሼͲιǡ ͳͷιǡ ͵Ͳιǡ Ͷͷιǡ Ͳιǡ ͷιǡ ͻͲιሽ. Thus, in total, 35 *.cae files were prepared and over 1000 simulations were carried out.
In Figure 17a and Figure 17b convergence for obtained values is presented. For SIFs and J-integral, the values from the third contour are enough stable. For T-stress convergence is slightly worse (cause of specific properties of this one), but is reliable from forth contour. Overall – average for 6 contours from 5 to 10 was calculated for all simulations as an output value.
(a)
(b) Figure 17. (a) The convergence of SIFs as a function of the contours number. Load case: ܽ ൌ ʹͲ݉݉ǡ ߠ ൌ ͵Ͳι ; (b) The convergence of J-integral and T-stress
parameters as a function of the contours number. Case: ܽ ൌ ʹͲ݉݉ǡ ߠ ൌ ͵Ͳι
As a confirmation of the author's script validity, in Figure 18a and Figure 18b several comparisons are presented including own elaborated results (KI/KII-BC1, KI/KII-BC2) and results obtained by Richard formulas (Eqs. 2 and 3) and values obtained from the simulation. According to the initial observations, the first type of boundary conditions (BC1)shows better predictions than the second (BC2) one, in particular for the SIF in mode II. Moreover, values obtained using BC1 conditions are comparable with values obtained by Richard’s values. However, for mode I both BCs provide excellent agreement with Richard’s SIFs values.
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(a)
(b)
Figure 18. Stress intensity factors for CTS specimen as a function of the load angle a: (a) Mode I comparison of BC1 and BC2 with Richard formula; (b) Mode II comparison of BC1 and BC2 with Richard formula Also, In Figure 19a, the T-stress as a function of crack tip location on the x-axis is presented as a result of the performed simulations. It is well known that high negative values of T-stress are a factor that is responsible for the stabilisation of the fracture. However, during mixed-mode fatigue crack propagation, T-stress values are decreasing gradually – the crack is less stable. Hence, due to above, there is a possibility to bifurcating or sudden rupture. When the load angle increase in CTS specimen, the higher values of initial T-stress are more distant from 0. This observation explains why the initial crack is the most stable under mode I loading type and the less stable under mode II loading type. This difference is neglected in the end phase of crack propagation [38,39]. In Figure 19b, the J - integral as a function of crack tip location on the x-axis is presented. The differences between the considered load cases are not so significant, but J-integral is slightly higher for the cases where is more dominant mode I.
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(a)
(b) Figure 19. Fracture mechanics parameters for different load cases as a function of the horizontal crack tip location: (a) Values of T-stress for different force load angles ߠ as a function of the crack tip location on the x-axis; (b) Values of J-integral for different load angles ߠ angles as a function of the crack tip location on the x-axis
On the other hand, in Figure 20a the dimensionless SIFs for mode I, dependent on the location of the crack tip on the x-axis (referred to the specimen width - W) for the different initial length (marked as len12 – length 12 mm, etc.) of the notch and crack, are presented. Similarly, in Figure 19b the geometric factor is presented versus expression based on W and ligament W-x. The expression was selected in accordance with the [35]. As it was assumed there, the proposed dependence should ௪
య మ
be linear. Thus, the dimensionless SIFs are linearly dependent on ቀ௪ି௫ቁ .
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(a)
(b) Figure 20. Nondimensionalized characteristic of SIFs for CTS specimen: (a) Dimensionless SIFs for mode I dependent on the location of the crack tip on the xaxis for the different initial length of the notch and crack; (b) Dimensionless SIF for mode I dependent on the location of the crack tip on the x-axis for the different initial length of the notch and crack. Finally, In Figure 21a crack paths for different force direction ߠ were plotted for the case with an initial length of crack and notch of 20 mm. In Figure 21b crack path for the different initial length of precrack and notch are also shown.
(a)
(b) Figure 21. (a) The crack path for different force direction ߠ. Case: a = 20 mm; (b)
The crack path for the different initial length of precrack and mechanical notch
According to the performed numerical analysis, it is possible to compare experimental and numerical predictions of the fatigue crack initiation angles. This comparison is presented in Figure 22.
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Figure 22. Crack initiation angles; experiment vs. simulation 4. Kinetic Fatigue Fracture Diagrams (KFFD) The proposed numerical analyses and methodology allow to apply it for the postexperimental data. After the experiments, the fracture mechanics parameters were estimated for selected points corresponding with the fatigue crack paths. Fatigue crack growth rate diagrams with separated mode I and mode II are shown in Figure 23. As it is noticeable, in each case, the crack growth rate was significantly higher for mode II.
(a)
(b) Figure 23. Fatigue crack growth rate diagrams for mode I and mode II (a) for load angle θ=30°; (b) for load angle θ=45°; From an engineering point of view, fatigue crack growth prediction under mixed-mode
loading conditions can be expressed using the Paris’ law [40]. In this case, it can be rewritten as: ݀ܽ ൌ ܥ ൫ȟܭ ൯ ݀ܰ
(10)
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Where: Ceq – equivalent C Paris’ law constant, DKeq – equivalent SIF range, meq – equivalent m-exponent from Paris’ law mode I. Equivalent SIF range can be calculated using different criteria, one of the most widely used is criterion proposed by Tanaka [41]: ర
ܭ ൌ ඨܭூସ ͺܭூூସ
ସ ͺܭூூூ ͳെݒ
(11)
In Equation 11 n represents the Poisson’s ratio. According to the Tanaka criterion [41], the generalized fatigue crack growth rate diagram has been constructed. All results are presented in Figure 24.
Figure 24. Generalised fatigue crack growth rate diagram for mode I and mixedmode (I+II) loading condition As it is expected, for all loading cases, mixed-mode fatigue crack growth curves are coincident with mode I result. There is also a small scattering of exponents ݉ and ܥ of the Paris’ law equation.
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5. Conclusions The paper presents experimental and numerical research on the fatigue crack growth in the pearlitic steel from the rail. In the experimental campaign, the CTS specimens were involved. For calculation of the fracture mechanics parameters, the comprehensive numerical analyses of the CTS specimen in a complex stress state were carried out. According to the combined experimental-numerical approach, the Kinetic Fatigue Fracture Diagrams were determined for the mode I loading as well as for the mixed modes of crack growth. Based on the obtained experimental and numerical results, the following conclusions may be drawn: ·
According to the numerical simulations, high initial mixity constraint decreases with increasing fatigue crack length.
·
The proposed numerical procedure has been successfully validated using stationary Richard solutions. The presented approach allows determining fatigue crack growth curves for each type of the mixed-mode (I+II) loading condition,
·
Fatigue lifetime of the CTS specimen increases with higher initial mixity level (load angle θ).
·
In rail steel, the fatigue crack growth rate under mode II condition was significantly higher than under mode I (see Fig. 23) for each loading angle θ.
·
Mixed mode (I+II) fatigue crack growth rate for tested rail steel can be successfully characterized using Tanaka criterion (Eq. 11).
Acknowledgements: This work was supported by grant number 2018/02/X/ST8/02041 (02NA/0001/19) financed by the Polish National Science Centre (Narodowe Centrum Nauki, NCN). References 1.
M. Kaneta, H. Yatsuzuka, Y. Murakami Mechanism of crack growth in lubricated rolling/sliding contact ASLE Trans, 28 (3) (1985), pp. 407-414
2.
M.R. Phillips, C.J.S. Chapman A magnetic method for detecting the onset of surface contact fatigue Wear, 49 (2) (Aug. 1978), pp. 265-272
3.
Nejad, R. M., Shariati, M., & Farhangdoost, K. (2019). Prediction of fatigue crack propagation and fractography of rail steel. Theoretical and Applied Fracture Mechanics, 101, 320-331.
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4.
Christodoulou, P. I., Kermanidis, A. T., & Haidemenopoulos, G. N. (2016). Fatigue and fracture behavior of pearlitic Grade 900A steel used in railway applications. Theoretical and Applied Fracture Mechanics, 83, 51-59.
5.
Dimaki, A.; Shilko, E.; Psakhie, S.; Popov, V. Simulation of fracture using a meshdependent fracture criterion in the discrete element method. Facta Univ. - Series Mech. Eng. 2018, 16, 1, 41-50.
6.
Erdogan, F., & Sih, G. C. (1963). On the crack extension in plates under plane loading and transverse shear.
7.
Nuismer, R.J. An energy release rate criterion for mixed mode fracture. Int. J. Fract. 1975, 11, 245–250.
8.
Sih, G.C. Strain-energy-density factor applied to mixed mode crack problems. Int. J. Fract. 1974, 10, 305–321.
9.
Makinde, A.; Thibodeau, L.; Neale, K.W. Development of an apparatus for biaxial testing using cruciform specimens. Exp. Mech. 1992, 32, 2, 138-144
10. Kibler, J.J.; Roberts, R. The Effect of Biaxial Stresses on Fatigue and Fracture. J. Eng. Ind. 1970, 92, 727. 11. Hannon, A.; Tiernan, P. A review of planar biaxial tensile test systems for sheet metal. J. Mater. Process. Technol. 2008, 198, 1-3, 1-13 12. Smits, A.; Van Hemelrijck, D.; Philippidis, T.P.; Cardon, A. Design of a cruciform specimen for biaxial testing of fibre reinforced composite laminates. Compos. Sci. Technol. 2006, 66, 7-8, 964-975 13. Breitbarth, E.; Besel, M.; Reh, S. Biaxial testing of cruciform specimens representing characteristics of a metallic airplane fuselage section. Int. J. Fatigue 2018, 108, 116-126 14. Abd-Elhady, A.A. Mixed mode I/II SIFs through the thickness of disc type specimens. Eng. Solid Mech. 2013, 1, 119–128. 15. Nobile, L. Mixed mode crack initiation and direction in beams with edge crack. Theor. Appl. Fract. Mech. 2000, 33, 2, 107-116 16. Yen, S. Ǧ C; Craddock, J.N.; Teh, K.T. Evaluation of a modified arcan fixture for the inǦ plane shear test of materials. Exp. Tech. 1988, 12, 12, 22-25
17. Greer, J.M.; Galyon Dorman, S.E.; Hammond, M.J. Some comments on the Arcan mixed-mode (I/II) test specimen. Eng. Fract. Mech. 2011, 78, 2088–2094. 18. Richard, H. A. (1984). Some theoretical and experimental aspects of mixed mode fractures. In Fracture 84 (pp. 3337-3344). Pergamon.
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19. Diekmann, P., Buchholz, F. G., Grebner, H., & Richard, H. A. (1991). J Integral Calculations in Mixed-Mode Elastic-Plastic Crack Problems. In EGF9. 20. Lesiuk, G.; Kucharski, P.; Correia, J.A.F.O.; De Jesus, A.M.P.; Rebelo, C.; Simões da Silva, L. Mixed mode (I+II) fatigue crack growth in puddle iron. Eng. Fract. Mech. 2017, 185, 175-192 21. Rozumek, D.; Marciniak, Z.; Lesiuk, G.; Correia, J.A.; de Jesus, A.M.P. Experimental and numerical investigation of mixed mode I + II and I + III fatigue crack growth in S355J0 steel. Int. J. Fatigue 2018, 113, 160-170 22. Ferreira, J.; Correia, J.A.F.O.; Lesiuk, G.; González, S.B.; Gonzalez, M.C.R.; De Jesus, A.M.P.; Fernández-Canteli, A. Pre-strain effects on mixed-mode fatigue crack propagation behaviour of the P355NL1 pressure vessels steel. In Proceedings of the American Society of Mechanical Engineers, Pressure Vessels and Piping Division (Publication) PVP; 2018; Vol. 6A-2018. 23. Monotonic and Ultra-Low-Cycle Fatigue Behaviour of Pipeline Steels; Fernandes, A.A., Jesus, A.M.P. de, Natal Jorge, R., Eds.; Springer International Publishing: Cham, 2018; ISBN 978-3-319-78095-5 DOI:10.1007/978-3-319-78096-2. 24. Albuquerque, C.; Silva, A.L.L.; De Jesus, A.M.P.; Calçada, R. An efficient methodology for fatigue damage assessment of bridge details using modal superposition of SIFs. Int. J. Fatigue 2015, 81, 61-77 25. Rozumek, D.; Lachowicz, C.T.; Macha, E. Analytical and numerical evaluation of SIF along crack paths in the cruciform specimens under out-of-phase cyclic loading. Eng. Fract. Mech. 2010, 77, 11, 1808-1821 26. Williams, M.L. On the Stress State at the Base of a Stationary Crack. J. Appl. Mech. 1957, 28, 1, 78-82 27. Irwin, G.R. Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate. In Proceedings of the Applied Mechanics Division Summer Conference, ASME; 1957, 24, IX, 361-364 28. Westergaard, H.M.W. Bearing pressures and cracks. J. Appl. Mech. 1939, 6, 49-53 29. Gupta, M.; Alderliesten, R.C.; Benedictus, R. A review of T-stress and its effects in fracture mechanics. Eng. Fract. Mech. 2015, 134, 218-241 30. Shlyannikov, V.N. T-stress for crack paths in test specimens subject to mixed mode loading. Eng. Fract. Mech. 2013, 108, 3-18 31. Seitl, S.; Knésl, Z. Two parameter fracture mechanics: Fatigue crack behavior under mixed mode conditions. Eng. Fract. Mech. 2008, 75, 3-4, 857-865
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32. Nazarali, Q.; Wang, X. The effect of T-stress on crack-tip plastic zones under mixedmode loading conditions. Fatigue Fract. Eng. Mater. Struct. 2011, 34, 10, 792-803 33. Richard, H.A. A new compact shear specimen. Int. J. Fract. 1981, 17, 5 34. Richard, H.A. Some Theoretical and Experimental Aspects of Mixed Mode Fractures. In Fracture 2013, 84, DOI:10.1016/b978-1-4832-8440-8.50358-6. 35. Antunes, F. V.; Branco, R.; Ferreira, J.A.M.; Borrego, L.P. SIF solutions for CTS mixed mode specimen. Frat. ed Integrità Strutt. 2019, 13, 676–692. 36. Simulia
Abaqus
Scripting
User
Guide
Available
online:
https://www.sharcnet.ca/Software/Abaqus613/. 37. Abaqus 6.13 Abaqus 6.13. Anal. User’s Guide. Dassault Syst. 2013. 38. Meliani, O.B.M.H.; Pluvinage, G. A review of T-stress calculation methods in fracture mechanics computation. Nat. Technol. 2016, 20–30. 39. Gupta, M.; Alderliesten, R.C.; Benedictus, R. A review of T-stress and its effects in fracture mechanics. Eng. Fract. Mech. 2015, 134, 218–241. 40. Paris, P. C.; Erdogan., F. A critical analysis of crack propagation laws. J. Basic Eng. 1963, 85.4: 528-533. 41. Tanaka K., Fatigue crack propagation from a crack inclined to the cyclic tensile axis. Eng. Fract. Mech. 1974, 6, 493-507.
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Highlights ·
The mixed mode (I+II) FCGR results for rail steel have been presented.
·
Fracture mechanics parameters for CTS specimen were determined.
·
Tanaka criterion was successfully used for FCGR description,
·
Fractography study was performed for different mixity level
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Declaration of interests ܈The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ܆The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S1350-6307(19)31640-1 https://doi.org/10.1016/j.engfailanal.2019.104354 EFA 104354
To appear in:
Engineering Failure Analysis
Received Date: Revised Date: Accepted Date:
4 November 2019 8 December 2019 16 December 2019
Please cite this article as: Lesiuk, G., Smolnicki, M., Mech, R., Zięty, A., Fragassa, C., Analysis of fatigue crack growth under mixed mode (I+II) loading conditions in rail steel using CTS specimen, Engineering Failure Analysis (2019), doi: https://doi.org/10.1016/j.engfailanal.2019.104354
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Analysis of fatigue crack growth under mixed mode (I+II) loading conditions in rail steel using CTS specimen
Grzegorz Lesiuk1*, Michał Smolnicki 1, Rafał Mech1, Anna Zięty1, Cristiano Fragassa2 1
Wroclaw University of Science and Technology, Faculty of Mechanical Engineering, Department of Mechanics, Materials Science and Engineering, Smoluchowskiego 25, PL50-370 Wrocław; Poland
2
University of Bologna, Department of Industrial Engineering, Viale Risorgimento 6, 40133 Bologna, Italy
* Corresponding Author: [email protected] [G.L.] Abstract: The paper is focused on the investigation of fatigue crack growth in rail steel. Mode I and its combination with mode II are one of the frequent reason for rail failures. Therefore, mode I and mixed-mode (I+II) experiments were carried out in order to obtain the fatigue crack growth curves for typical rail steel. In experimental campaign were used CTS (Compact Tension Shear) specimens. The Stress Intensity Factors (SIF) were determined for all load cases using Finite Element Method (FEM) with developed and automated Authors’ tool ABAQUS environment. During the experimental and numerical study, the impact of different elastic mixity level on fatigue crack growth rates and fatigue crack growth directions has been discussed. As it is noticeable the decrease of the mode I domination (under mixed-mode loading condition) caused an increase of the specimen fatigue lifetime. This observation was also discussed in the light of the detailed fatigue crack paths study using Scanning Electron Microscope (SEM). Keywords: fatigue fracture mechanics, fatigue crack path study, mixed-mode I+II fatigue crack growth, CTS, SIF, FEM, rail steel
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1. Introduction The safety of a steel rail structure is often determined by the maintenance of the rail system and the prediction of its failure based on the fracture process. The mechanics of train wheel-rail movement causes a specific configuration of stress state in rail structure so-called Rolling Contact Fatigue (RCF). The cyclic nature of loading combined with multiaxial stress state in rail determines the specific crack initiation point and fatigue crack paths during cyclic loading. The nature of the type of crack morphology is well reported in the works [1-2]. On the other hand, the fatigue crack growth properties are still required in several numerical approaches in fatigue lifetime prediction of structural elements and components like rails. Also, fatigue crack growth rate (FCGR) parameters are often one of the major criteria for evaluation of the rail steel quality (i.e. pearlitic, bainitic etc.). Recently, Nejad et al. [3] and Christodoulou [4] studied the fatigue crack growth behaviour under mode I of rail steel 900A regarding the influence of stress ratios R=0.1, 0.2, 0.3. However, in some cases, the mixedmode (I+II) fatigue crack behaviour is required in fatigue analysis of the structural integrity. Therefore the Authors decided to investigate the same class 900A rail steel behaviour under mixed-mode (I+II) in order to investigate real behaviour of mixed-mode fatigue crack growth in rail steel class 900A. As it is well known, in some cases, due to material flaws and heterogeneity with a combination of the complex stress state, cracks propagate under mixed-mode loading conditions. The problem needs an individual approach to research work in comparison to the standard problem of one mode fracture, which is already well elaborated. Most of the theoretical works are focused on crack initiation and propagation criteria [5-8]. Between many others, are noteworthy: Maximum Tangential Stress (MTS) [6], Maximum Energy Release Rate (MERR)[7] and Minimum Strain Energy Density (SED)[8]. For the validation of such criteria, multiaxial loading tests are required, as widely described in [9-11] with mode I+II using cruciform specimens [9, 12, 13]. From the practical point of view, the mixed-mode fracture mechanics is mostly studied using uniaxial tensile machines. At present, there are different types of specimens, which can be used to test fracture under mixed-mode (I+II) loading using the uniaxial tensile machine. For brittle materials are often used disc-type specimens such as Centre Cracked Circular Disc (CCCD) [11] and Semi-Circular Bend (SCB) [14]. Other popular types of specimens are Single Edge Crack (SEC)[15], Single Edge Notched Beam[15]. Another group of specimens is based on the extended concept of CT (Compact Tension) specimen; Arcan (“butterfly-shaped”) [16, 17] and CTS (CTS) [18, 19] specimens. As it is observed, this type of specimen is most frequently used in material
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characterisation under mixed-mode (I+II) condition. For static load conditions, SIF (SIF) solutions are available in the literature [20]. However, some of the significant problems in engineering practice are related to fatigue and fatigue fracture. In those cases, the closed-form SIF solutions are invalid [21-22]. As an alternative, a FEM (FEM) methodology can be used for the evaluation of SIFs [23-26]. Based on this, the numerical support of the experimental part is an essential contribution for fatigue crack growth analysis under mixed-mode loading condition of rail steel. According to the proposed methodology and obtained experimental results, it can help avoid critical situations (using a damage tolerance approach) of railway lines. 2. Material and Methods 2.1. Material investigations As an object of Authors’ interest, steel coming from the rail line (profile UIC60) was selected. For the tests, the unserviceable rail was taken directly from the manufacturer. Samples for examination were cut out under cooling with liquid in order to avoid microstructural changes. The microscopic observations were performed using light microscopy (LM) in the non-etched and etched state (5%HNO3). The microstructure of the tested steel is shown in Figure 1. The chemical composition of the material was spectrally analysed. The results are summarized in Table 1. Table 1. Chemical composition (in % by weight) of delivered rail Material
C
Mn
Si
P
S
Cr
Ni
Mo
Fe
tested rail steel
0.721
0.873
0.256
0.012
0.005
0.053
0.032
0.011
bal.
Static properties (UTS – ultimate tensile strength, spl – yield stress, A5 – elongation at break, HV – Vickers Hardness) for selected material are collected in Table 2 (mean value of five results). The static tensile test was performed following the PN-EN ISO 6892-1:201609 standard using round specimen with diameter Ød=6mm extracted from the rail. Stressstrain curves collected from the static tensile test are presented in Figure 2. According to the chemical composition, the microstructure of the tested material corresponds to the perlite structure of steel, typical for railway rails.
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Table 2. Static mechanical properties of analysed metallic materials Material Extracted rail steel
UTS
spl/s0.2
A5
HV
(MPa)
(MPa)
(%)
(-)
998
481
14.5
258
Figure 1. The pearlitic microstructure of the extracted material from UIC60 rail; a) representative image with non-metallic inclusions chains in the non-etched state (200x) b) typical microstructure of tested steel (200x), c) enlarged pearlite structure with ferrite grains (400x), d) typical fatigue crack path in rail steel (100x)
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1000
(MPa)
800 600 400 200 0 0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
Figure 2. Monotonic stress-strain curves for tested rail steel 2.2. Mode I and mixed-mode (I+II) fatigue crack growth – theoretical and experimental aspects In 1957 Williams [27], proposed an asymptotic expansion of the stress field in the area surrounding the tip of the sharp crack. Original formula (Equation 1), reporting the stresses ߪ in the function of distance from the crack tip r, Young Modulus E, Poisson’s coefficient ߥǡstrains ߜ ǡand angle η, is presented below: ߪ ൌ
ܭூ
ξʹߨݎ
݂ூ ሺߟሻ
ܭூூ
ξʹߨݎ
݂ூூ ሺߟሻ
ܭூூூ
ξʹߨݎ
݂ூூூ ሺߟሻ ܶߜଵ ߜଵ
(1)
ሺ ܶݒ ߝܧଷଷ ሻߜଷ ߜଷ ܱ൫ξݎ൯
As is it well known, that here exist SIFs for three modes: KI, KII, KIII as well as on quantity T – called T – stresses. The latter one is the transverse component of stress and was first proposed by Irwin [28] to deal with problems using Westergaard solution [29] to a uniaxial type of loading. Gupta et al. [30] gathered information about the influence of the T-stress parameter. This approach is also reflected in various approaches in mixed mode fatigue fracture description as in [31-33]. However, T-stress influence is still investigated and needs intensively study on in several types of specimens in order to standardise testing procedures. On the other hand, during mode I and mixed mode-fatigue crack growth, the most accepted approach is based on the DK approach. It is also preferred in most widely used American Standard - ASTM E647 devoted to the description of testing method, but only for mode I. In case of the mixed-mode loading type (I+II) the CTS Specimen is most widely used.
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This specimen was designed and investigated by Richard [18, 19, 33, 34]. It is worth to note that this specimen needs an additional fixture for clamping on the uniaxial tensile machine. The general concept and an exemplary measurement stand is shown in Figure 3.
Figure 3. CTS specimen mixed-mode loading configuration; (a) general concept; (b) definition of θ- angle; (c) experimental measurement stand; 1 – load cell, 2 – grips, 3 – specimen holder, 4 – CTS specimen, 5, 6 - lightening system, 7 stereoscopic microscope, 8 – digital camera By changing the load direction θ it is possible to create different mixed-mode loading conditions; for ߠ ൌ Ͳιis pure mode I (so ܭூூ ൌ Ͳ), for ߠ ൌ ͻͲι - pure mode II (so ܭூ ൌ Ͳ) and between these values of θ, there are mixed-mode loading conditions. Figure 4 presents
the main dimensions of CTS specimen [15]. Richard proposed closed-form solutions for SIFs using fundamental geometrical aspects of CTS specimen like a – crack length, W – specimen width, t – specimen thickness (see Equation 2 and Equation 3) [34]. ܽ ͲǤʹ ʹǤͷ ቀܹ െ ܽቁ ܨή ξߨܽ ή ߠ ܭூ ൌ ଶ ܽ ඩ ܹ ݐቀͳ െ ܹ ቁ ͳ ͲǤͷͷ ቀ ܽ ቁ െ ͲǤͺͺ ቀ ܽ ቁ ܹെܽ ܹെܽ
ܽ െͲǤʹ͵ ͳǤͶ ቀܹ െ ܽቁ ܨή ξߨܽ ή ߠ ܭூூ ൌ ଶ ܽ ඩ ܹ ݐቀͳ െ ቁ ͳ െ ͲǤ ቀ ܽ ቁ െ ʹǤͲͺ ቀ ܽ ቁ ܹ ܹെܽ ܹെܽ
(2)
(3)
However, the cited equations are only valid for stationary cracks.
This type of CTS-specimen is also suitable for cyclic loading. In the experimental campaign was used MTS 809 servo-hydraulic machine (Figure 3) equipped with a digital camera for capturing the crack tip on the CTS specimen (schematically drawn in Figure 4). Three different load angles were considered θ = 0°, 30°,45°. Before fatigue crack growth test,
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the precracking procedure was involved by cyclic loading with ܨ௫ ൌ ͷǤͷ݇ܰ and ܴ ൌ
ͲǤͳduring 150,000 cycles and with ୫ୟ୶ ൌ ͶǤͷ and ൌ ͲǤͳ during remaining 50,000
cycles. After precracking procedure, the test was performed using a constant load amplitude method. During experiment Fmax=8 kN, f=10 Hz and R=0.1 were kept.
Figure 4. Geometric scheme of the exemplary CTS specimen used in the experimental and numerical campaign (all dimensions in mm, thickness t=8mm) In order to estimate fatigue crack growth behaviour under the mode I and mixed-mode (I+II), the kinetic fatigue fracture diagrams (KFFD) were constructed. Directly from the experiment, the fatigue crack lengths and number of cycles to failure were registered. Using an automated digital imaging system (Figure 3), the crack tip was detected and recorded in the local coordinate system. It is located at the end of precrack (mode I) length – Figure 4. Fatigue crack growth curves are shown in Figure 5.
Figure 4. Exemplary CTS specimen after the test (load angle 45 degrees) with real crack path and the definition of the coordinate system for numerical analysis
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Figure 5. Fatigue crack growth curves for tested rail steel After fatigue experiments, the mixed-mode (I+II) crack initiation angles ψ0 were measured using the microscope. All results are presented in Figure 6.
Figure 6. Crack initiation angles in CTS specimens after fatigue crack growth test The detailed images of fracture surface for mode I and mixed-mode (I+II, load angle 45 degrees) are presented in Figs 7-10. For mode I, the fracture surface is mostly shaped by the typical mode I, the dominant tensile mechanism of fracture. It is noticeable in enlarged microscopic view (Figure 7) the numerous secondary cracks characteristic for higher mode I, tensile stress level. This transgranular type of fracture also shows pearlite colonies and lamellar spacing in steel grains. This type of fracture is dominant along the fatigue crack path. Only last, overloaded part (Figure 8) of the specimen is shaped by brittle, transgranular facets with secondary cracks.
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Figure 7. Initial crack path and fracture surface (mode I) of tested material – macro view and marked by frame view on the fracture surface
Figure 8. Final crack path (3mm) and overloaded fracture surface (mode I) of tested material– macro view and marked by frame view on the fracture surface The fracture surface of the specimen subjected to mixed-mode (I+II) loading type is different – Figure 8. At the initial region, here is a noticeable sliding type of fatigue crack growth (magnified view) – it might be caused by relatively high KII/KI ratio on the initial crack length. However, in the last stage of fatigue crack growth (Figure 10), it is a noticeable dominant mode I, tensile type of fatigue crack growth mechanism, similar to pure mode I.
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Figure 9. Initial crack path and fracture surface (mode I+II, loading angle 45 deg.) of tested material – macro view and marked by frame view on the fracture surface
Figure 10. Final crack path (3mm) and overloaded fracture surface (mode I+II, loading angle 45 deg.) of tested material– macro view and marked by frame view on the fracture surface 3. Numerical study of the CTS specimen – SIF estimation In order to construct fatigue crack growth rate diagrams, it is necessary to calculate SIFs. The SIFs solutions provided by Richard are valid only for stationary cracks, which is reported in papers of Richard [33,34] and proven in many applications in [20,21]. On the other hand, there is a strong need to elaborate generalised model in a closed-form solution for CTS specimen most widely used in fatigue fracture mechanics. According to the generalised form of SIF solution, dimensionless quantities YI, YII are used below as independent from of force value and stress level: ܻூ ൌ
ܭூ
ɐඥߨܽ
ൌ
ܭூ ή ܹݐ
ή ɐඥߨܽ
(4)
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ܻூூ ൌ
ܭூூ
ɐඥߨܽ
ൌ
ܭூூ ή ܹݐ
ή ɐඥߨܽ
(5)
In the classic approach where the stationary crack is investigated aeff means crack length which is equal to the projection of this crack on the x-axis. However, during fracture under mixed-mode loading, crack tends to deviate from a straight line. Thus, two possible values of aeff are considered: total length of notch depth, precrack and crack as well as their projection on the x-axis. 3.1. Automatic generation of the numerical model Since the multiplicity and the relevance of parameters affecting the tests in the case of mixed-mode loading conditions, automation in the simulation process is strictly necessary for efficient use of the CTS specimen method, For instance, even dimensionless ܻூ ǡ ܻூூ still
depends on a few different quantities.
In work [35] authors using Buckingham π theorem show that this is possible to reduce ௫
௬
the set of quantities defining dimensionless SIFs to ቄௐ ǡ ௐ ǡ ߚǡ ߙ െ ߚቅwhere ߙ – crack angle and ߚ – force direction angle. Of course, there are other sets that can fulfil the same function. ௫
Due to the character of the prepared script in this work set ሼௐ ǡ ௐ ǡ ߙǡ ߚ } is used. This
approach is more complicated for finding dependencies for a specific location of the crack tip, but is more natural, when using with experimental research.
Simulia Abaqus environment allows taking advantage of scripting with the reduction of the analysis complexity. Based on Abaqus Scripting User Guide [36] the Abaqus Scripting Interface is an application programming interface (API), which gives the possibility to access to the models and data operating by Abaqus. The Abaqus Scripting Interface is based on Python language, thus it is possible to interact with Abaqus objects by using Python scripts. The User has different possibilities, inter alia, ·
creation and changing Abaqus object i.e. parts, sections, materials, loads, boundary conditions and steps,
·
management of Abaqus jobs, including submit,
·
management of Abaqus output database. Using Abaqus Scripting Interface a script to simulate fracture in CTS specimen was
prepared. In Figure 11 flowchart of this script is presented.
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Figure 11. Flowchart of the script for fatigue fracture analysis of CTS specimen First, input from the user about specimen geometry is required, crack parameters and initial notch/precrack. Based on this information, the geometry model is created (eventually included data from the previous iteration, if present), then crack is included in the model via partitioning. In the next few steps, the numerical model is created i.e. material, assembly, boundary condition, finite element mesh and history output are defined. In the purpose of obtaining fracture data crack region is defined by using the Contour Integral Method. Finally, the work submission is automatically realized and, after results are provided, data necessary to calculate the new position of crack and fracture parameters are obtained. Part of the data also serves as input for the next iteration. The whole process is continued until the specimen is not fully broken into two parts. Result of one run of presented script is a set of models and result files for the subsequent increments of the crack for the user-defined geometry and other initial values like force action direction. In addition to the above text file with aggregated data is generated. Using loops, it
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is possible to automatically generate a model for a user-defined list of parameters i.e. for 12 different initial precrack lengths. Consequently, the whole procedure of obtaining requested data is automatized. According to the main goal of this paper, the CTS specimen was numerically investigated in order to extract essential fracture mechanics parameters for FCGR description in rail steel. In Figure 12, a scheme of the specimen used in the experimental and numerical campaign is presented. The main dimensions of this type of specimen are expressed by: the length of precrack – ܽǡ the spacing of pins –ܾ, thickness ݐǡ and width ݓare previously
used in some formulas for calculating SIFs and will be discussed in the latter part of this work.
Figure 12. Scheme of CTS specimen: length of precrack – a , pins spacing – b, thickness t, and width W For calculating SIFs the Contour Integral method (as detailed later in the article) was involved. Material data specified in Abaqus simulations was following; Young Modulus ܧൌ ʹͳͳͲͲͲܽܲܯand Poisson’s ratio ߥ ൌ ͲǤ͵Ǥ
Several different ways to apply boundary conditions are proposed in the literature in the
case of CTS specimens. In Figure 13a (later referenced as Boundary Conditions type 1 – BC1) and Figure 13b (later referenced as Boundary Conditions type 2 – BC2), two popular methods are presented [37-39]. In the first case, the supports on the bottom part of the model generate reaction forces which are equal to forces applied in these points in the second case. These two propositions were compared and checked via preliminary FEM simulations. In the presented study, the Richard [34] formula (cf. Equation 2 and Equation 3) was used to evaluate these results in the basic case when there is no crack propagation.
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Comparison of SIFs for the standard dimension (as described in subchapter geometric model) and ߠ ൌ ͵Ͳιis shown below in Table 3.
(a)
(b)
Figure 13. (a) Boundary conditions type 1. Forces on one side of the specimen and fixing on the other side; (b) Boundary conditions type 2. Forces on both sides of the specimen. Table 3. Comparison between analytical solutions and results from different types of BC BC – type I ܭூ ሾ ܽܲܯή ݉݉Ǥହ ]
ܭூூ ሾ ܽܲܯή ݉݉Ǥହ ]
400.53 -116.64
Richard
BC – type II ȟ ൌ ͲǤ͵Ψ ȟ ൌ ͳǤͷΨ
405.06 -72.79
Solution ȟ ൌ ͳǤͶͻΨ
ȟ ൌ ͵ͺǤΨ
399.11 -118.73
According to data presented in Table 3. first type of boundary conditions was selected. Forces ܨଵ ǡ ܨଶ ǡ ܨଷ were calculated based on two conditions: first - the resultant force has the
desirable direction of action ߠ and second -resultant momentum of forces is equal 0, so there
is no bending. Forces ܨଵ ǡ ܨଶ ܽ݊݀ܨଷ which meet these requirements were described in equations 6, 7 and 8. On the other side of specimen degree of freedom was taken as it is shown in Figure 13a. ܿ ܨଵ ൌ ܨή ቀͲǤͷ ߙ ߙቁ ܾ ܨଶ ൌ ܨή ߙ
(6) (7)
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ܿ ܨଷ ൌ ܨή ቀͲǤͷ ߙ െ ߙቁ ܾ
(8)
Due to two-dimensional modelling space, the force applied to the specimen ܨmust be
normalised via division by the thickness of the specimen ܨௗ ൌ ܨȀݐ. Both forces and boundary conditions are applied to the pins represented by reference points in Abaqus. Kinematic coupling (all degree of freedom) is used to model these conditions. Discrete model, as well as previous elements of the numerical model, is generated in the automatized process. The size of the elements is one of the parameters which can be regulated. The influence of the element size on the obtained exemplary SIF values for CTS specimen is shown in Fig. 14.
Figure 14. KI and KII values vs. finite element size – load angle 45°, mixed-mode fatigue crack length 5 mm According to the exemplary data in Fig. 14, it is noticeable, that for the smaller than 0.5 mm finite element sizes, there is a significant change in the obtained SIF values. In the presented case, the mesh size estimated on the level about 0.5 mm corresponds approximately to a 1/80 specimen width. During simulations, mostly linear quadrilateral type CPS4R (4node bilinear plane stress quadrilateral with reduced integration as well as hourglass control) were used. The general view of the discrete model is presented in Figure 15a. Regarding the fact that the J-integral and T-stresses are calculated, regular circular layers of finite elements around the crack tip were applied - Figure 15b. In another region, the more coarse mesh has been applied.
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(b) (a) Figure 15. A discrete model of the CTS specimen (a) Mesh on the whole specimen, size of element circa equal width of a specimen divided by 80; (b) Mesh in the crack region - 10 contours for calculating SIFs, J-Integral and T-stress There are few different ways to obtain SIFs, J-Integral and T-stresses in Abaqus environments such as eXtended FEM and Contour Integral. First of them can be easily applied in more complicated cases. In the case of the two-dimensional model, the second of them is preferred. This procedure may cause problems with the presence of a singularity near the crack tip. In order to overcome this, a standard procedure was applied. Nodes at the crack tip were translated by 0.25 of element length. Besides, the collapsed element side (single node) method was selected for dealing with the degenerate element. Here, it is required to deal with the problem of singularity in the tip of the sharp crack [40]. The direction of the crack extension was defined based on the crack increase from the previous step (or based on the direction of precrack for the first simulation). The exemplary definition is shown in Figure 16.
Figure 16. Definition of the crack extension direction
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3.2. Results of numerical simulations To determine the influence of carried
out
for
the
ௐ
following
ratio as well as force direction ߠ simulations were distance
of
notch
and
precrack:
ܽ א
ሼͳʹǡ ͳǡ ʹͲǡ ʹͶǡ ʹͺǡ ͵ʹሽand following force direction: ߠ אሼͲιǡ ͳͷιǡ ͵Ͳιǡ Ͷͷιǡ Ͳιǡ ͷιǡ ͻͲιሽ. Thus, in total, 35 *.cae files were prepared and over 1000 simulations were carried out.
In Figure 17a and Figure 17b convergence for obtained values is presented. For SIFs and J-integral, the values from the third contour are enough stable. For T-stress convergence is slightly worse (cause of specific properties of this one), but is reliable from forth contour. Overall – average for 6 contours from 5 to 10 was calculated for all simulations as an output value.
(a)
(b) Figure 17. (a) The convergence of SIFs as a function of the contours number. Load case: ܽ ൌ ʹͲ݉݉ǡ ߠ ൌ ͵Ͳι ; (b) The convergence of J-integral and T-stress
parameters as a function of the contours number. Case: ܽ ൌ ʹͲ݉݉ǡ ߠ ൌ ͵Ͳι
As a confirmation of the author's script validity, in Figure 18a and Figure 18b several comparisons are presented including own elaborated results (KI/KII-BC1, KI/KII-BC2) and results obtained by Richard formulas (Eqs. 2 and 3) and values obtained from the simulation. According to the initial observations, the first type of boundary conditions (BC1)shows better predictions than the second (BC2) one, in particular for the SIF in mode II. Moreover, values obtained using BC1 conditions are comparable with values obtained by Richard’s values. However, for mode I both BCs provide excellent agreement with Richard’s SIFs values.
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(a)
(b)
Figure 18. Stress intensity factors for CTS specimen as a function of the load angle a: (a) Mode I comparison of BC1 and BC2 with Richard formula; (b) Mode II comparison of BC1 and BC2 with Richard formula Also, In Figure 19a, the T-stress as a function of crack tip location on the x-axis is presented as a result of the performed simulations. It is well known that high negative values of T-stress are a factor that is responsible for the stabilisation of the fracture. However, during mixed-mode fatigue crack propagation, T-stress values are decreasing gradually – the crack is less stable. Hence, due to above, there is a possibility to bifurcating or sudden rupture. When the load angle increase in CTS specimen, the higher values of initial T-stress are more distant from 0. This observation explains why the initial crack is the most stable under mode I loading type and the less stable under mode II loading type. This difference is neglected in the end phase of crack propagation [38,39]. In Figure 19b, the J - integral as a function of crack tip location on the x-axis is presented. The differences between the considered load cases are not so significant, but J-integral is slightly higher for the cases where is more dominant mode I.
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(a)
(b) Figure 19. Fracture mechanics parameters for different load cases as a function of the horizontal crack tip location: (a) Values of T-stress for different force load angles ߠ as a function of the crack tip location on the x-axis; (b) Values of J-integral for different load angles ߠ angles as a function of the crack tip location on the x-axis
On the other hand, in Figure 20a the dimensionless SIFs for mode I, dependent on the location of the crack tip on the x-axis (referred to the specimen width - W) for the different initial length (marked as len12 – length 12 mm, etc.) of the notch and crack, are presented. Similarly, in Figure 19b the geometric factor is presented versus expression based on W and ligament W-x. The expression was selected in accordance with the [35]. As it was assumed there, the proposed dependence should ௪
య మ
be linear. Thus, the dimensionless SIFs are linearly dependent on ቀ௪ି௫ቁ .
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(a)
(b) Figure 20. Nondimensionalized characteristic of SIFs for CTS specimen: (a) Dimensionless SIFs for mode I dependent on the location of the crack tip on the xaxis for the different initial length of the notch and crack; (b) Dimensionless SIF for mode I dependent on the location of the crack tip on the x-axis for the different initial length of the notch and crack. Finally, In Figure 21a crack paths for different force direction ߠ were plotted for the case with an initial length of crack and notch of 20 mm. In Figure 21b crack path for the different initial length of precrack and notch are also shown.
(a)
(b) Figure 21. (a) The crack path for different force direction ߠ. Case: a = 20 mm; (b)
The crack path for the different initial length of precrack and mechanical notch
According to the performed numerical analysis, it is possible to compare experimental and numerical predictions of the fatigue crack initiation angles. This comparison is presented in Figure 22.
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Figure 22. Crack initiation angles; experiment vs. simulation 4. Kinetic Fatigue Fracture Diagrams (KFFD) The proposed numerical analyses and methodology allow to apply it for the postexperimental data. After the experiments, the fracture mechanics parameters were estimated for selected points corresponding with the fatigue crack paths. Fatigue crack growth rate diagrams with separated mode I and mode II are shown in Figure 23. As it is noticeable, in each case, the crack growth rate was significantly higher for mode II.
(a)
(b) Figure 23. Fatigue crack growth rate diagrams for mode I and mode II (a) for load angle θ=30°; (b) for load angle θ=45°; From an engineering point of view, fatigue crack growth prediction under mixed-mode
loading conditions can be expressed using the Paris’ law [40]. In this case, it can be rewritten as: ݀ܽ ൌ ܥ ൫ȟܭ ൯ ݀ܰ
(10)
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Where: Ceq – equivalent C Paris’ law constant, DKeq – equivalent SIF range, meq – equivalent m-exponent from Paris’ law mode I. Equivalent SIF range can be calculated using different criteria, one of the most widely used is criterion proposed by Tanaka [41]: ర
ܭ ൌ ඨܭூସ ͺܭூூସ
ସ ͺܭூூூ ͳെݒ
(11)
In Equation 11 n represents the Poisson’s ratio. According to the Tanaka criterion [41], the generalized fatigue crack growth rate diagram has been constructed. All results are presented in Figure 24.
Figure 24. Generalised fatigue crack growth rate diagram for mode I and mixedmode (I+II) loading condition As it is expected, for all loading cases, mixed-mode fatigue crack growth curves are coincident with mode I result. There is also a small scattering of exponents ݉ and ܥ of the Paris’ law equation.
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5. Conclusions The paper presents experimental and numerical research on the fatigue crack growth in the pearlitic steel from the rail. In the experimental campaign, the CTS specimens were involved. For calculation of the fracture mechanics parameters, the comprehensive numerical analyses of the CTS specimen in a complex stress state were carried out. According to the combined experimental-numerical approach, the Kinetic Fatigue Fracture Diagrams were determined for the mode I loading as well as for the mixed modes of crack growth. Based on the obtained experimental and numerical results, the following conclusions may be drawn: ·
According to the numerical simulations, high initial mixity constraint decreases with increasing fatigue crack length.
·
The proposed numerical procedure has been successfully validated using stationary Richard solutions. The presented approach allows determining fatigue crack growth curves for each type of the mixed-mode (I+II) loading condition,
·
Fatigue lifetime of the CTS specimen increases with higher initial mixity level (load angle θ).
·
In rail steel, the fatigue crack growth rate under mode II condition was significantly higher than under mode I (see Fig. 23) for each loading angle θ.
·
Mixed mode (I+II) fatigue crack growth rate for tested rail steel can be successfully characterized using Tanaka criterion (Eq. 11).
Acknowledgements: This work was supported by grant number 2018/02/X/ST8/02041 (02NA/0001/19) financed by the Polish National Science Centre (Narodowe Centrum Nauki, NCN). References 1.
M. Kaneta, H. Yatsuzuka, Y. Murakami Mechanism of crack growth in lubricated rolling/sliding contact ASLE Trans, 28 (3) (1985), pp. 407-414
2.
M.R. Phillips, C.J.S. Chapman A magnetic method for detecting the onset of surface contact fatigue Wear, 49 (2) (Aug. 1978), pp. 265-272
3.
Nejad, R. M., Shariati, M., & Farhangdoost, K. (2019). Prediction of fatigue crack propagation and fractography of rail steel. Theoretical and Applied Fracture Mechanics, 101, 320-331.
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4.
Christodoulou, P. I., Kermanidis, A. T., & Haidemenopoulos, G. N. (2016). Fatigue and fracture behavior of pearlitic Grade 900A steel used in railway applications. Theoretical and Applied Fracture Mechanics, 83, 51-59.
5.
Dimaki, A.; Shilko, E.; Psakhie, S.; Popov, V. Simulation of fracture using a meshdependent fracture criterion in the discrete element method. Facta Univ. - Series Mech. Eng. 2018, 16, 1, 41-50.
6.
Erdogan, F., & Sih, G. C. (1963). On the crack extension in plates under plane loading and transverse shear.
7.
Nuismer, R.J. An energy release rate criterion for mixed mode fracture. Int. J. Fract. 1975, 11, 245–250.
8.
Sih, G.C. Strain-energy-density factor applied to mixed mode crack problems. Int. J. Fract. 1974, 10, 305–321.
9.
Makinde, A.; Thibodeau, L.; Neale, K.W. Development of an apparatus for biaxial testing using cruciform specimens. Exp. Mech. 1992, 32, 2, 138-144
10. Kibler, J.J.; Roberts, R. The Effect of Biaxial Stresses on Fatigue and Fracture. J. Eng. Ind. 1970, 92, 727. 11. Hannon, A.; Tiernan, P. A review of planar biaxial tensile test systems for sheet metal. J. Mater. Process. Technol. 2008, 198, 1-3, 1-13 12. Smits, A.; Van Hemelrijck, D.; Philippidis, T.P.; Cardon, A. Design of a cruciform specimen for biaxial testing of fibre reinforced composite laminates. Compos. Sci. Technol. 2006, 66, 7-8, 964-975 13. Breitbarth, E.; Besel, M.; Reh, S. Biaxial testing of cruciform specimens representing characteristics of a metallic airplane fuselage section. Int. J. Fatigue 2018, 108, 116-126 14. Abd-Elhady, A.A. Mixed mode I/II SIFs through the thickness of disc type specimens. Eng. Solid Mech. 2013, 1, 119–128. 15. Nobile, L. Mixed mode crack initiation and direction in beams with edge crack. Theor. Appl. Fract. Mech. 2000, 33, 2, 107-116 16. Yen, S. Ǧ C; Craddock, J.N.; Teh, K.T. Evaluation of a modified arcan fixture for the inǦ plane shear test of materials. Exp. Tech. 1988, 12, 12, 22-25
17. Greer, J.M.; Galyon Dorman, S.E.; Hammond, M.J. Some comments on the Arcan mixed-mode (I/II) test specimen. Eng. Fract. Mech. 2011, 78, 2088–2094. 18. Richard, H. A. (1984). Some theoretical and experimental aspects of mixed mode fractures. In Fracture 84 (pp. 3337-3344). Pergamon.
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19. Diekmann, P., Buchholz, F. G., Grebner, H., & Richard, H. A. (1991). J Integral Calculations in Mixed-Mode Elastic-Plastic Crack Problems. In EGF9. 20. Lesiuk, G.; Kucharski, P.; Correia, J.A.F.O.; De Jesus, A.M.P.; Rebelo, C.; Simões da Silva, L. Mixed mode (I+II) fatigue crack growth in puddle iron. Eng. Fract. Mech. 2017, 185, 175-192 21. Rozumek, D.; Marciniak, Z.; Lesiuk, G.; Correia, J.A.; de Jesus, A.M.P. Experimental and numerical investigation of mixed mode I + II and I + III fatigue crack growth in S355J0 steel. Int. J. Fatigue 2018, 113, 160-170 22. Ferreira, J.; Correia, J.A.F.O.; Lesiuk, G.; González, S.B.; Gonzalez, M.C.R.; De Jesus, A.M.P.; Fernández-Canteli, A. Pre-strain effects on mixed-mode fatigue crack propagation behaviour of the P355NL1 pressure vessels steel. In Proceedings of the American Society of Mechanical Engineers, Pressure Vessels and Piping Division (Publication) PVP; 2018; Vol. 6A-2018. 23. Monotonic and Ultra-Low-Cycle Fatigue Behaviour of Pipeline Steels; Fernandes, A.A., Jesus, A.M.P. de, Natal Jorge, R., Eds.; Springer International Publishing: Cham, 2018; ISBN 978-3-319-78095-5 DOI:10.1007/978-3-319-78096-2. 24. Albuquerque, C.; Silva, A.L.L.; De Jesus, A.M.P.; Calçada, R. An efficient methodology for fatigue damage assessment of bridge details using modal superposition of SIFs. Int. J. Fatigue 2015, 81, 61-77 25. Rozumek, D.; Lachowicz, C.T.; Macha, E. Analytical and numerical evaluation of SIF along crack paths in the cruciform specimens under out-of-phase cyclic loading. Eng. Fract. Mech. 2010, 77, 11, 1808-1821 26. Williams, M.L. On the Stress State at the Base of a Stationary Crack. J. Appl. Mech. 1957, 28, 1, 78-82 27. Irwin, G.R. Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate. In Proceedings of the Applied Mechanics Division Summer Conference, ASME; 1957, 24, IX, 361-364 28. Westergaard, H.M.W. Bearing pressures and cracks. J. Appl. Mech. 1939, 6, 49-53 29. Gupta, M.; Alderliesten, R.C.; Benedictus, R. A review of T-stress and its effects in fracture mechanics. Eng. Fract. Mech. 2015, 134, 218-241 30. Shlyannikov, V.N. T-stress for crack paths in test specimens subject to mixed mode loading. Eng. Fract. Mech. 2013, 108, 3-18 31. Seitl, S.; Knésl, Z. Two parameter fracture mechanics: Fatigue crack behavior under mixed mode conditions. Eng. Fract. Mech. 2008, 75, 3-4, 857-865
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32. Nazarali, Q.; Wang, X. The effect of T-stress on crack-tip plastic zones under mixedmode loading conditions. Fatigue Fract. Eng. Mater. Struct. 2011, 34, 10, 792-803 33. Richard, H.A. A new compact shear specimen. Int. J. Fract. 1981, 17, 5 34. Richard, H.A. Some Theoretical and Experimental Aspects of Mixed Mode Fractures. In Fracture 2013, 84, DOI:10.1016/b978-1-4832-8440-8.50358-6. 35. Antunes, F. V.; Branco, R.; Ferreira, J.A.M.; Borrego, L.P. SIF solutions for CTS mixed mode specimen. Frat. ed Integrità Strutt. 2019, 13, 676–692. 36. Simulia
Abaqus
Scripting
User
Guide
Available
online:
https://www.sharcnet.ca/Software/Abaqus613/. 37. Abaqus 6.13 Abaqus 6.13. Anal. User’s Guide. Dassault Syst. 2013. 38. Meliani, O.B.M.H.; Pluvinage, G. A review of T-stress calculation methods in fracture mechanics computation. Nat. Technol. 2016, 20–30. 39. Gupta, M.; Alderliesten, R.C.; Benedictus, R. A review of T-stress and its effects in fracture mechanics. Eng. Fract. Mech. 2015, 134, 218–241. 40. Paris, P. C.; Erdogan., F. A critical analysis of crack propagation laws. J. Basic Eng. 1963, 85.4: 528-533. 41. Tanaka K., Fatigue crack propagation from a crack inclined to the cyclic tensile axis. Eng. Fract. Mech. 1974, 6, 493-507.
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Highlights ·
The mixed mode (I+II) FCGR results for rail steel have been presented.
·
Fracture mechanics parameters for CTS specimen were determined.
·
Tanaka criterion was successfully used for FCGR description,
·
Fractography study was performed for different mixity level
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Declaration of interests ܈The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ܆The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: